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Asymptotic behavior of solutions of some difference equations defined by weakly dependent random vectors. (English) Zbl 1333.39014

The authors are motivated by the recent paper of K.-I. Yoshihara [Yokohama Math. J. 58, 1–15 (2012; Zbl 1278.60099)], where the relationship between the Black-Scholes-type differential equation \(dX(t)=\sigma X(t)\,dW(t)+\nu X(t)\,dt\) and its discrete counterpart \[ \Delta X(t_k)=X(t_k)-X(t_{k-1})=X(t_{k-1})\left[\frac{\nu T}{n}+ \sqrt{\frac{T}{n}}\sigma \xi_k\right] \] is investigated. The present paper extends this result in two directions: (i) a pair of more general equations \[ dX(t)=\,X(t)\left[h(t)dt+v(t)\,dW(t)\right], \] and \[ \Delta X(t_k)=\,X(t_{k-1})\left[h(t_{k-1})\frac{T}{n}+ \sqrt{\frac{T}{n}}v(t_{k-1})\xi_k\right] \tag{*} \] is considered, (ii) the quantities \(X,\xi\) are allowed to be multidimensional.
Additional conditions are formulated under which the quantity \(S^{(n)}(T)\) for the solution \[ S^{(n)}(t)=z\exp\left\{\sum_{k=1}^n \log\left(1+h(t_{k-1}) \frac{T}{n}+\sqrt{\frac{T}{n}}v(t_{k-1})\xi_k\right)\right\} \] of (*) converges almost surely to \[ x(T)=z\exp\left\{\int_0^T h(t)\,dt-\frac{1}{2}\int_0^T v^2(t)\,dt+\gamma_1 \int_0^T v(t)\,dW(t)\right\}. \] Some extension of this result to a more general pair of equations is also formulated.

MSC:

39A50 Stochastic difference equations
39A30 Stability theory for difference equations
60G10 Stationary stochastic processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
39A12 Discrete version of topics in analysis

Citations:

Zbl 1278.60099
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References:

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